Arrow's theorem is either meaningless or false

 In this post I will show that if Arrow's Theorem is meaningful, Arrow's theorem must be false for two different reasons.

Firstly, a preference aggregation mechanism can have the form- 'If Continuum Hypothesis is false then resolve cylicity issues in such and such manner'. Arrow's theorem says that this mechanism can't meet certain conditions. But it is impossible to be sure because we don't know for certain if CH is false. In other words, we can say nothing a priori about a function which can't currently be evaluated. The fact is, we know of no type of mathematics of sufficient complexity to be useful which is also 'complete'. Thus we can't say anything about certain functions which can be formulated in that mathematical language but which we can neither prove nor disprove. 

A different problem has to do with distinguishability of alternatives. The theorem states that no rank-order electoral system can be designed that always satisfies three "fairness" criteria. However a 'rank-order electoral system' can only be designed where alternatives can be clearly distinguished in an objective manner. If this is not the case, the theorem is meaningless. Now, it is certainly true that we can easily distinguish the different 'candidates' for whom our Society can vote. But 'candidates' aren't truly mutually exclusive social 'alternatives'. Thus, if a particular candidate promises to fulfil the agenda of a different candidate, and puts forward reasons why she will be more effective in this regard, then some people will change their ranking.

To give an example, when Tony Blair, in 1995, scrapped Labour's old Clause 4 re. common ownership of industry, he could be seen as more Thatcherite than the ineffectual John Major. Rupert Murdoch gave him his imprimatur. Suddenly the Labour candidate was more Tory than the Tory candidate. 

 More generally, the social alternatives represented by the candidates changes as the knowledge base changes.  True, one could arbitrarily stipulate that only mutually exclusive alternatives are discernible for the electoral system. But then one could also arbitrarily stipulate that any two alternatives are actually one and the same, or actually the opposite of what they were, and thus no 'cyclicity' can obtain. After all, a society or a species has a different 'fitness landscape' from any individual or even the set of all individuals. It is that landscape which promotes or demotes candidates from the ranks of those fit to live.

A similar problem arises with personal 'preferences'. One could arbitrarily stipulate that agent x has preferences y which are independent of the knowledge base or the preferences of others. But one could as plausibly stipulate that x's preferences are merely an instrument designed attain quite different preferences. Indeed, a society may be a mechanism for its participants to attain this end- i.e. we are part of a Society only to the extent that we wish our own impulses and idiosyncratic affective states are replaced by what is canonical for our Society.  Thus, when I chose to become a 'naturalised' British citizen, I did so because I wanted to give up certain prejudices, or unmeaning affiliations, of my own so as to embrace what was 'natural', or non-arbitrary, or canonical, for willing and devoted subjects of the British Crown in Parliament. No doubt, the fact that remaining an Indian citizen meant being a subject of a corrupt, incompetent, anti-Hindu, Italian dynasty influenced my decision. 

 My point is preferences, and meta-preferences, too are epistemic and impredicative. There is no objective ranking associated with each individual. Thus there is nothing objective, save by arbitrary stipulation, which is aggregated by a voting procedure. All we can say is that such decisions have a degree of legitimacy for purely historical or hysteresis based reasons. But there is no ergodicity- nothing 'Economic' in Samuelson's sense, here. 

Thus, either Arrow's theorem is meaningless because there is nothing objective which is being aggregated nor is aggregation itself actually a rank order rather than an arbitrary procedure, or else, if by some magic, we have access to the preferences people would have 'at the end of mathematical time' when everything becomes known and the most complex mathematical calculations have all been completed- in other words, all problems of impredicativity have been solved and every intension has a well defined extension, then it must also be the case that the voting procedure that is used is also one that everybody thinks is as fair as possible- or fair 'naturaliter'. 

This is because one of the things a Society can choose is how it objectively defines preferences and how it objectively aggregates them. To argue otherwise- viz. that no Socieity can determine what is objective- is to suggest that noting can be 'sui generis'. Yet such an argument is itself sui generis. This is like Witlessstein's objection to Russell's ramified Type theory. At the end of mathematical time, there can be a ramified Type theory which has naturality. There could even be 'atomic propositions' precisely because a 'multiple relation' theory would correspond to a 'carving up of the world along its joints'. 

Notice, from that vantage point, the three fairness conditions Arrow mentions must also be fulfilled. We may say 'what is the deterministic method by which this is done?' The answer is 'we will only find out at the end of mathematical time when every non-deterministic procedure has a deterministic counterpart'.

 Alternatively, we could say that it will happen when every 'lawless' choice sequence is discovered to be law-like. It may still be argued that subjectively it is true that any accessible voting procedure is arbitrary and unfair. But, it is equally true that, equally subjectively, we don't know that it isn't providential and the best possible outcome under the circumstances from some higher point of view. There is no objective way to say one view is superior or more 'natural' or non-arbitrary than the other. 

Suppose we have a voting procedure to rank the natural numbers. It may be that the majority of people think the number 6 is cute and sweet and should be higher than the number 9. But if 9 is relegated to the sixth place while 6 is pushed forward to the ninth place all that has really happened is that the symbols for 6 and 9 have been swapped around. Thus what you have is still the same ranking of the natural numbers albeit with different symbols in different places. What if there is 'cyclicity'? People want 6 to be higher than 9 and 9 to be higher than 8 but 8 to be higher than 6? The answer is that there is no new ranking of the natural numbers. People just use the symbols used for the natural numbers in other countries or by guys doing useful work. They may come to the conclusion that it is silly to put such matters to the vote. My point is that voting procedures can be used for silly purposes but lessons from such silliness can be quickly learned. In 1897, the Indiana Legislature was about to pass a bill defining the value of pi such that the circle could be squared. Luckily, a Maths professor happened to be present and got the Senate to quash the bill by pointing out that, under the Bill, pi would have 4 different values. But even if it had been passed, all that would have happened was that Indiana would have been laughed at. Its own Maths teachers would have ignored the law. Still, in the event, a lesson was learned. Legislatures became wary of voting for Bills lobbied for by crackpots. 

Arrow's theorem was the work of a crackpot who didn't understand that in Economics only correlation, not causation, is discernible. This is what we call 'Granger causality'. At the 'end of mathematical time', sure, you could have a transitive ordering because the correct way of 'carving up the world along its joints' had been found and all Granger causality had become actual causality. Here the 'characteristics space' would coincide with the Social 'configuration space' because all states of the world would be known and so no Knightian uncertainty would obtain. But, by then, all mathematics would have ended. The human race and the Universe itself would, of course, have ended much previously. Still at that point Arrow's theorem would be meaningful but false because preferences would be whatever they would have to be for aggregation to be Pareto optimal, non-dictatorial, and feature truly independent alternatives irrelevant to rank ordering. Moreover, it would only be at that point that the universal domain would be accessible.